- Email: [email protected]
Effect of volumetric absorption coefficient on Rosseland radiative heat transfer
ht.
J. He,,r Mers
Transfer.
Vol. 10, pp. 1049-1054.
Pergamon
Press Ltd.
1967.
Pnnted
in Great
Britain
R. L. SPICER The RAND Corporation,
Santa Monica, California
(Received 5 March 1965 and in revisedform I November 1966)
Abstract-In calculating the temperature profile and heat transfer associated with an optically dense fluid it has been the practice to use Rosseland’s approximation for the radiative heat flux, since its form lends itself well for use in partial differential equations. However, very little is known about the form of the absorption coefficient which is the fluid-characteristic term in the approximation. It is revealed in this analysis that in the case of uniform fluid (optically dense) flow past a flat plate, the calculated temperature profiles and heat transfer to the wall by radiation alone are sensitive to the choice of the form of the absorption coefficient.
NOMENCLATURE CP,
f, h, k 6 4, T,
u, U,
u, %
Y*
specific heat ; dependent variable (7’) ; specific enthalpy ; volumetric absorption coefficient ; power parameter ; radiation flux ; absolute temperature; fluid velocity ; fluid velocity parallel to wall (U); fluid velociety perpendicular to wall (zero) ; distance along wall ; perpendicular distance from wall.
Greek symbols similarity variable ; 92 fluid density ; P? Stefan-Boltzmann constant. 6,
or dense gas receives radiative energy by photo absorption only from points very close to it. Therefore, for such a gas, which is also molecularly dense (number of collisions is large enough to maintain a Maxwellian distribution of excited states corresponding to the local temperature) local thermodynamic equilibrium exists ; i.e. emission from this infinitesimal volume is the same as it would be from a black body in equilibrium at the same temperature. With the assumptions of an optically dense gas, local thermodynamic equilibrium, and isotropic emission, an integration over all directions of the conservation of radiation energy equation yields the Rosseland approximation for the radiation flux vector [l-3]. For the one-dimensional heat-transfer case, the Rosseland equation becomes : - 160T3 dT qR
Subscripts power parameter ; 4 radiation; R, wall ; W, free stream. CQ, INTRODUCTION
AN INFINITESIMALvolume
in an optically
thick
=
3k,
dy ’
The purpose of this paper is to determine the effect of the volumetric absorption coefficient k, on equilibrium radiative heat transfer qR in an optically thick fluid, where the Rosseland approximation for qR is used. To accomplish this, it is assumed that kR is proportional to the 1049
R.
2050
L.
SPICER
absolute temperature to the nth power. This assumption is taken for mathematical convenience; however, it does represent a qualitatively correct trend, and does allow an investigation of the effect of kR on qR through the use of n as a parameter. THEORETICAL
assuming a constant cP the following relation is obtained. 2
eng
= (3 - n)T’2-“’
aT (
%J
+
*w.zz:
>
$v2
(3) where
CONCEPTS
The flow considered herein is an inviscid incompressible fluid passing the edge of a flat plate at a constant velocity U,. The flow enters at a uniform temperature lb, and the plate temperature is held constant at T, (see Fig. 1).
FIG. 1.
For this parallel flow, the conservation equations (energy and momentum) are uncoupled, and the conservation of mass and momentum equations, for the steady-state case, yield the trivial solution, u = U, and u = 0. Now, by assuming that the wail radiates just as the gas does, and neglecting heat transfer by conduction, the energy equation becomes [4]
with the boundary conditions being y = 0,
T = T,,
y = co,
T = T,,
x = 0,
T = T,.
(4)
Equation (3) is a nonlinear second order partial differential equation which has no closed form solution. However, equations of this form have been thoroughly investigated by several authors such as Boltzmann, Crank, and others in determining solutions to other physical problems. With the given boundary conditions, equation (3) can be transformed into a nonlinear ordinary differential equation by the use of the Boltzmann transformation or similarity variable. The form of this key transformation or similarity variable is clearly given in the works of Crank 1[6] and Heaslet and Alksne [S]. which are concerned with the time dependent diffusion problem. In noting these works it is readily determined that the similarity variable, in this case, is given by
where dh = c,dT. Using the Rosseland approximation for an optically thick fluid and incorporating k, = aT” = (k,,/T”,)T” - 16oT@ -‘) dT qR
By
substituting
=
-
3a
-dy.
(2)
equation (1) into equation (2),
where r, =
16aT”, 3~, u, cp &co’
This indeed, allows equation (3) to be transformed into the following nonlinear ordinary differential equation.
EFFECT
f”f’3-n)
OF
+ (3 - n)f’2-“) (j-y
VOLUMETRIC
+ 2q,f
ABSORPTION
= 0 (5)
with the following two boundary conditions f(0)
= TW
(6)
f(a)
= T,.
(7)
It is noted here that when n = 3, equation (5) reduces to the linear case which was solved by Goulard [4] wherein the normalized solution for the temperature distribution reduces to the Gauss Error Function [7] in terms of the similarity variable. NUMERICAL
INTERPRETATION
paper is to show the effect of n on both the temperature profiles near the wall and the heat transfer to the wall. Therefore, before any meaningful comparisons of the results can be obtained from the solutions to the differential equation (5). the dependence of ye. on n must be removed, since 9. represents a scaled distance from the wall and the scale varies with n. This dependence on n can be removed by redefining the independent variable as ‘I=‘I.T;~.
Therefore comes
OF RESULTS
At this point it should be noted that the independent variable chosen (similarity variable) is dependent upon the value of n. The intent of this
(8)
the new independent
variable
be-
(9)
rl = (2&x)
SOLUTIONS
Equation (5) with the boundary conditions given by equations (6) and (7) has no closed form solution. It represents a two-point boundary value problem which may be solved by a series expansion method as in [S] or by one of several numerical integration techniques. Reference [8] displays several numerical methods of solving linear ordinary differential equations ; however, these methods as such, do not guarantee that a convergence process exists when nonlinearities are present, as they are in equation (5). Therefore, by authors choice, a technique of trial and error using the calculus of finite difference was adopted. Equation (5) in finite differences form, has been programmed in Fortran for use on the IBM 7044 digital computer, such that desired solutions (f as a tabular function of q. and n) were obtained. Since numbers had to be used in this approach, representative values of T, and T, were chosen as 10000°K and 500°K respectively. These numbers were chosen since air at extremely high temperatures [2] (associated with super-orbital re-entry and blast wave flow fields) becomes optically dense.
1051
COEFFICIENT
where 160
r=
3~, u, cp k,,’ This change in the independent variable was performed and the solutions (temperature vs. r]) to equation (5) for n = 0, 1, 2, 3, 4, 5, and 6 are displayed in Fig. 2. In order to get a better understanding of these results, Fig. 3 is given. This figure is a replot of Fig. 2; it shows the temperature profiles near the wall, for various n’s Here q can be considered as a scaled distance from the wall (in the y direction) at some value of x, where these profiles vary with l! Jx along the wall. Note that the larger the value of n is, the cooler the flow is near the wall since more heat is being transferred to the wall. HEAT
TRANSFER
AT THE WALL
The main purpose herein is to determine the effect of the absorption coefficient through the parameter n, on heat transfer, qR, at the wall (at y = 0 + r] = 0). In order to show the effect of the parameter n on qR, recall equation (2): - 16aTC3-“‘i3T qR
=
3a
-.
8Y
R. L. SPICER
1052
with computations were performed numerical tua the independent variable. but here, the dependence on n has been removed and 1 is pwen as the
5
5
104
5
IO3
Similarity
5 variable,
I06 7
5
IO’
5
IOE
(Ko3$1
FIG. 2. Numerical solutions to the differential equation (5) for various values of the parameter n.
In terms of the similarity variable equation (2) becomes
108 5
Cc dT
qR -=
T(“-3)
cx
d,,
(10)
where 2’
5
z .z P ’ = g E e
cx =
-8a
3k,,($X)
106
Therefore at the wall : 5(11)
u” s Ix a
I05
Equation (11) shows the relative relationship between the heat transfer to the wall, and temperature and temperature gradient at the wall.
5
0 01 0 z I04 5
I03 I02
i (7;,,I) Temperature,
T(OKl
FIG. 3. Temperatire profiles for various values of n.
Figure 4 displays the temperature gradient at q = 0 vs. the power n. This figure shows a considerable variation in (dT,/dn),=o with n. The initial slope increases very sharply as n decreases. Table 1 shows the total effect of n on qRW [through the use of equation (ll)]. Note, as n increases by a value of one, the magnitude of qRW/cxincreases by as much as a factor of 3.28,
EFFECT
OF VOLUMETRIC
ABSORPTION
COEFFICIENT
1053
by the process of ru~iatio~ than do the small values of n. This effect is to be expected since as n increases, the absorption coeff’cient, kR, decreases (T/‘&, c 1); therefore less energy is absorbed by the cooler portion of the flow and more energy is radiated to the wall. CONCLUSIONS
is seen from Figs. 3 and 4 that the temperature profile and wall temperature gradient are very much dependent upon the value of the parameter n. Table 1 indicates that the mag~tude of the net energy radiated to the wall also varies with n; however, the choice of n is not as critical as the temperature gradient dependence infers. Therefore, in working a problem in radiation where the relationship k, N T” is used, the degree of the precision with which n is chosen should depend on which flow quantity is of interest. It
n(ND)
FICZ.4. Temperature gradient at wall vs. the parameter n.
Table 1. Radiative heat transfer at the wall
REFERENCES equations of radiation gas dynamics, Purdue University A&ES 62-4, p. 50 (March 1962). R. GOULARD,Similarity parameters in radiation gas dynamics, Purdue University A&ES 62-8, p. 32 (September 1962). R. VISKANTA,Reat transfer in thermal radiation absorbine and scattering media, ANL-6170, p. 87 (May 1960). R. GOULARD,The transition from black body to Rosseland formulations in optically thick flows, Int. J. Heat MUSS Transfer 6 (lo), 873 (1963). M. A. HEASLET and A. ALKSNE, Diffusion from a fixed surface with a concentrat~on~e~ndent coefficient, J. Sot. I&. Appl. Math. 9 (4), 588 (1961). J. CRA+~X,The ~at~e~t~cs ofl)$iiion, pp. 148and 162. Clarendon Press, Oxford (1956). M. JAKOE, Heat Transfer, Vol. I, p. 252. Wiley and Sons, New York (1957). L. Fox, numerics Solution of Two Point Boundury Problems. Clarendon Press, Oxford (1957).
1. R. GOULARD,Fundamental
(> qRW
n
CX
0 1 2
3.500 x IO’ I.940 x 10” 1.328 x lo-’
4.375 x IO9 4.850 x IO9 6GO x 109
:
1.071 x 1‘100 x 1.469 x 2.406 x
1.071 x 2,200 x 5.875 x 1.925 x
5 6
10-a 10-a 1O-4 lo-”
10’0 10’0 10’0 10’”
2.
3. 4. 5.
6.
for the range of n’s considered. From n = 0 to t? = 6, the energy transferred to the wall increases by a factor of forty-four, The large values of n yield more heat transfer to the wall
7. 8.
R&mm&-Dans le calcul du pro@ de tem~ratu~ et du transport de chaleur associe 8 un fluide optiquement dense, on a couramment utilisC l’approximation de Rosseland pour ie flux de chalenr par rayonnement, puisque sa forme la rend toute indiqu&e pour l’emploi dans da equations aux d&iv&es partielles. Cependant. on connalt trb peu la forme du coefficient d’absorption qui est le terme caracttsristique du fluide dans l’approximation. La theorie montre que dam le MS dun Ccoulement uniforme du fluide (optiquement dense) le long d’une plaque plane, les protils de temperature calcuRs et le transport de chaleur a la paroi le myonnement seul sont sensibles au choix de la forme du coefficient ~ab~~tion. Z~mmenfasstmg-Bei der Berechnung des Temperaturprofils und des Wllrmetiberganges in einem optisch dichten Medium ist es ilblich. die Roseland-~~erung fib die Strahlung zu ve~~deR da sich 3x
1054
R.
L. SPICER
ihre Form in die partielle Differentialgleichung ftigt. Jedoch ist sehr wenig iiber die Form des Absorptionskoehizienten bekannt, der in der Niiherung den fii das Medium charakteristischen Ausdruck darstellt. Es zeigt sich in dieser Analyse, dass im Fall einheitlicher Striimung eines (optisch dichten) Mediums entlang einer ebenen Platte die errechneten Temperaturprotile und der Wlrmetibergang an die Wande durch Strahlung allein, auf die Wahl der Form des Absorptionskoefftzienten emphndlich reagieren.
AHHOTB~JI--IJ 06bIIqHOtinpaKTMKe
paWeTa paCIIpe/JeJIeHWI TeMIIepaTypbI M TenJIOO6MeHa IS OnTAqeCKN IIJIOTHOtiCpene IlpMMeHReTCR npn6nmKeHHe POCCeJIaHfia AJIH JIyWCTOrO nOTOK: Tenna, T.K. ero @lopMa yRo6Ha RJIR peruemfl ~H@#lepeHIwanbHbIx ypaBHeHHti B '43CTHbIX IIpOH3BOfiHbIX. OnHaKO AaHHbIX 0 KOL'@jjI4~MeHTe a6cop6I&m, BXOAR~eM B aIInpOKCHMa~MI0. B BIlAeWIeHa,XapaKTepH3yIO~erO H(H~KOCTb,He~OCTaTO~HO.~ pe3yJIbTaTe aHaJIM3a BbIRCHMJIOCb, YTO B CJIyWe 06TeKaHMR IIJIOCKOt IIJI3CTHHbI OAHOpORHOti HCIIAKOCTbIO (OI1TB~eCliIl nnoTaoti)pac4eTmIe npo@am TemepaTypbI A TerIJIOO6MeHa Hanyqemebf Ha CT~HK~:~BHMC~~T OT Rbl6paHHOti ~OpMblKO3@I$lUI~lleHT3 a6cop6qm.
J. He,,r Mers
Transfer.
Vol. 10, pp. 1049-1054.
Pergamon
Press Ltd.
1967.
Pnnted
in Great
Britain
R. L. SPICER The RAND Corporation,
Santa Monica, California
(Received 5 March 1965 and in revisedform I November 1966)
Abstract-In calculating the temperature profile and heat transfer associated with an optically dense fluid it has been the practice to use Rosseland’s approximation for the radiative heat flux, since its form lends itself well for use in partial differential equations. However, very little is known about the form of the absorption coefficient which is the fluid-characteristic term in the approximation. It is revealed in this analysis that in the case of uniform fluid (optically dense) flow past a flat plate, the calculated temperature profiles and heat transfer to the wall by radiation alone are sensitive to the choice of the form of the absorption coefficient.
NOMENCLATURE CP,
f, h, k 6 4, T,
u, U,
u, %
Y*
specific heat ; dependent variable (7’) ; specific enthalpy ; volumetric absorption coefficient ; power parameter ; radiation flux ; absolute temperature; fluid velocity ; fluid velocity parallel to wall (U); fluid velociety perpendicular to wall (zero) ; distance along wall ; perpendicular distance from wall.
Greek symbols similarity variable ; 92 fluid density ; P? Stefan-Boltzmann constant. 6,
or dense gas receives radiative energy by photo absorption only from points very close to it. Therefore, for such a gas, which is also molecularly dense (number of collisions is large enough to maintain a Maxwellian distribution of excited states corresponding to the local temperature) local thermodynamic equilibrium exists ; i.e. emission from this infinitesimal volume is the same as it would be from a black body in equilibrium at the same temperature. With the assumptions of an optically dense gas, local thermodynamic equilibrium, and isotropic emission, an integration over all directions of the conservation of radiation energy equation yields the Rosseland approximation for the radiation flux vector [l-3]. For the one-dimensional heat-transfer case, the Rosseland equation becomes : - 160T3 dT qR
Subscripts power parameter ; 4 radiation; R, wall ; W, free stream. CQ, INTRODUCTION
AN INFINITESIMALvolume
in an optically
thick
=
3k,
dy ’
The purpose of this paper is to determine the effect of the volumetric absorption coefficient k, on equilibrium radiative heat transfer qR in an optically thick fluid, where the Rosseland approximation for qR is used. To accomplish this, it is assumed that kR is proportional to the 1049
R.
2050
L.
SPICER
absolute temperature to the nth power. This assumption is taken for mathematical convenience; however, it does represent a qualitatively correct trend, and does allow an investigation of the effect of kR on qR through the use of n as a parameter. THEORETICAL
assuming a constant cP the following relation is obtained. 2
eng
= (3 - n)T’2-“’
aT (
%J
+
*w.zz:
>
$v2
(3) where
CONCEPTS
The flow considered herein is an inviscid incompressible fluid passing the edge of a flat plate at a constant velocity U,. The flow enters at a uniform temperature lb, and the plate temperature is held constant at T, (see Fig. 1).
FIG. 1.
For this parallel flow, the conservation equations (energy and momentum) are uncoupled, and the conservation of mass and momentum equations, for the steady-state case, yield the trivial solution, u = U, and u = 0. Now, by assuming that the wail radiates just as the gas does, and neglecting heat transfer by conduction, the energy equation becomes [4]
with the boundary conditions being y = 0,
T = T,,
y = co,
T = T,,
x = 0,
T = T,.
(4)
Equation (3) is a nonlinear second order partial differential equation which has no closed form solution. However, equations of this form have been thoroughly investigated by several authors such as Boltzmann, Crank, and others in determining solutions to other physical problems. With the given boundary conditions, equation (3) can be transformed into a nonlinear ordinary differential equation by the use of the Boltzmann transformation or similarity variable. The form of this key transformation or similarity variable is clearly given in the works of Crank 1[6] and Heaslet and Alksne [S]. which are concerned with the time dependent diffusion problem. In noting these works it is readily determined that the similarity variable, in this case, is given by
where dh = c,dT. Using the Rosseland approximation for an optically thick fluid and incorporating k, = aT” = (k,,/T”,)T” - 16oT@ -‘) dT qR
By
substituting
=
-
3a
-dy.
(2)
equation (1) into equation (2),
where r, =
16aT”, 3~, u, cp &co’
This indeed, allows equation (3) to be transformed into the following nonlinear ordinary differential equation.
EFFECT
f”f’3-n)
OF
+ (3 - n)f’2-“) (j-y
VOLUMETRIC
+ 2q,f
ABSORPTION
= 0 (5)
with the following two boundary conditions f(0)
= TW
(6)
f(a)
= T,.
(7)
It is noted here that when n = 3, equation (5) reduces to the linear case which was solved by Goulard [4] wherein the normalized solution for the temperature distribution reduces to the Gauss Error Function [7] in terms of the similarity variable. NUMERICAL
INTERPRETATION
paper is to show the effect of n on both the temperature profiles near the wall and the heat transfer to the wall. Therefore, before any meaningful comparisons of the results can be obtained from the solutions to the differential equation (5). the dependence of ye. on n must be removed, since 9. represents a scaled distance from the wall and the scale varies with n. This dependence on n can be removed by redefining the independent variable as ‘I=‘I.T;~.
Therefore comes
OF RESULTS
At this point it should be noted that the independent variable chosen (similarity variable) is dependent upon the value of n. The intent of this
(8)
the new independent
variable
be-
(9)
rl = (2&x)
SOLUTIONS
Equation (5) with the boundary conditions given by equations (6) and (7) has no closed form solution. It represents a two-point boundary value problem which may be solved by a series expansion method as in [S] or by one of several numerical integration techniques. Reference [8] displays several numerical methods of solving linear ordinary differential equations ; however, these methods as such, do not guarantee that a convergence process exists when nonlinearities are present, as they are in equation (5). Therefore, by authors choice, a technique of trial and error using the calculus of finite difference was adopted. Equation (5) in finite differences form, has been programmed in Fortran for use on the IBM 7044 digital computer, such that desired solutions (f as a tabular function of q. and n) were obtained. Since numbers had to be used in this approach, representative values of T, and T, were chosen as 10000°K and 500°K respectively. These numbers were chosen since air at extremely high temperatures [2] (associated with super-orbital re-entry and blast wave flow fields) becomes optically dense.
1051
COEFFICIENT
where 160
r=
3~, u, cp k,,’ This change in the independent variable was performed and the solutions (temperature vs. r]) to equation (5) for n = 0, 1, 2, 3, 4, 5, and 6 are displayed in Fig. 2. In order to get a better understanding of these results, Fig. 3 is given. This figure is a replot of Fig. 2; it shows the temperature profiles near the wall, for various n’s Here q can be considered as a scaled distance from the wall (in the y direction) at some value of x, where these profiles vary with l! Jx along the wall. Note that the larger the value of n is, the cooler the flow is near the wall since more heat is being transferred to the wall. HEAT
TRANSFER
AT THE WALL
The main purpose herein is to determine the effect of the absorption coefficient through the parameter n, on heat transfer, qR, at the wall (at y = 0 + r] = 0). In order to show the effect of the parameter n on qR, recall equation (2): - 16aTC3-“‘i3T qR
=
3a
-.
8Y
R. L. SPICER
1052
with computations were performed numerical tua the independent variable. but here, the dependence on n has been removed and 1 is pwen as the
5
5
104
5
IO3
Similarity
5 variable,
I06 7
5
IO’
5
IOE
(Ko3$1
FIG. 2. Numerical solutions to the differential equation (5) for various values of the parameter n.
In terms of the similarity variable equation (2) becomes
108 5
Cc dT
qR -=
T(“-3)
cx
d,,
(10)
where 2’
5
z .z P ’ = g E e
cx =
-8a
3k,,($X)
106
Therefore at the wall : 5(11)
u” s Ix a
I05
Equation (11) shows the relative relationship between the heat transfer to the wall, and temperature and temperature gradient at the wall.
5
0 01 0 z I04 5
I03 I02
i (7;,,I) Temperature,
T(OKl
FIG. 3. Temperatire profiles for various values of n.
Figure 4 displays the temperature gradient at q = 0 vs. the power n. This figure shows a considerable variation in (dT,/dn),=o with n. The initial slope increases very sharply as n decreases. Table 1 shows the total effect of n on qRW [through the use of equation (ll)]. Note, as n increases by a value of one, the magnitude of qRW/cxincreases by as much as a factor of 3.28,
EFFECT
OF VOLUMETRIC
ABSORPTION
COEFFICIENT
1053
by the process of ru~iatio~ than do the small values of n. This effect is to be expected since as n increases, the absorption coeff’cient, kR, decreases (T/‘&, c 1); therefore less energy is absorbed by the cooler portion of the flow and more energy is radiated to the wall. CONCLUSIONS
is seen from Figs. 3 and 4 that the temperature profile and wall temperature gradient are very much dependent upon the value of the parameter n. Table 1 indicates that the mag~tude of the net energy radiated to the wall also varies with n; however, the choice of n is not as critical as the temperature gradient dependence infers. Therefore, in working a problem in radiation where the relationship k, N T” is used, the degree of the precision with which n is chosen should depend on which flow quantity is of interest. It
n(ND)
FICZ.4. Temperature gradient at wall vs. the parameter n.
Table 1. Radiative heat transfer at the wall
REFERENCES equations of radiation gas dynamics, Purdue University A&ES 62-4, p. 50 (March 1962). R. GOULARD,Similarity parameters in radiation gas dynamics, Purdue University A&ES 62-8, p. 32 (September 1962). R. VISKANTA,Reat transfer in thermal radiation absorbine and scattering media, ANL-6170, p. 87 (May 1960). R. GOULARD,The transition from black body to Rosseland formulations in optically thick flows, Int. J. Heat MUSS Transfer 6 (lo), 873 (1963). M. A. HEASLET and A. ALKSNE, Diffusion from a fixed surface with a concentrat~on~e~ndent coefficient, J. Sot. I&. Appl. Math. 9 (4), 588 (1961). J. CRA+~X,The ~at~e~t~cs ofl)$iiion, pp. 148and 162. Clarendon Press, Oxford (1956). M. JAKOE, Heat Transfer, Vol. I, p. 252. Wiley and Sons, New York (1957). L. Fox, numerics Solution of Two Point Boundury Problems. Clarendon Press, Oxford (1957).
1. R. GOULARD,Fundamental
(> qRW
n
CX
0 1 2
3.500 x IO’ I.940 x 10” 1.328 x lo-’
4.375 x IO9 4.850 x IO9 6GO x 109
:
1.071 x 1‘100 x 1.469 x 2.406 x
1.071 x 2,200 x 5.875 x 1.925 x
5 6
10-a 10-a 1O-4 lo-”
10’0 10’0 10’0 10’”
2.
3. 4. 5.
6.
for the range of n’s considered. From n = 0 to t? = 6, the energy transferred to the wall increases by a factor of forty-four, The large values of n yield more heat transfer to the wall
7. 8.
R&mm&-Dans le calcul du pro@ de tem~ratu~ et du transport de chaleur associe 8 un fluide optiquement dense, on a couramment utilisC l’approximation de Rosseland pour ie flux de chalenr par rayonnement, puisque sa forme la rend toute indiqu&e pour l’emploi dans da equations aux d&iv&es partielles. Cependant. on connalt trb peu la forme du coefficient d’absorption qui est le terme caracttsristique du fluide dans l’approximation. La theorie montre que dam le MS dun Ccoulement uniforme du fluide (optiquement dense) le long d’une plaque plane, les protils de temperature calcuRs et le transport de chaleur a la paroi le myonnement seul sont sensibles au choix de la forme du coefficient ~ab~~tion. Z~mmenfasstmg-Bei der Berechnung des Temperaturprofils und des Wllrmetiberganges in einem optisch dichten Medium ist es ilblich. die Roseland-~~erung fib die Strahlung zu ve~~deR da sich 3x
1054
R.
L. SPICER
ihre Form in die partielle Differentialgleichung ftigt. Jedoch ist sehr wenig iiber die Form des Absorptionskoehizienten bekannt, der in der Niiherung den fii das Medium charakteristischen Ausdruck darstellt. Es zeigt sich in dieser Analyse, dass im Fall einheitlicher Striimung eines (optisch dichten) Mediums entlang einer ebenen Platte die errechneten Temperaturprotile und der Wlrmetibergang an die Wande durch Strahlung allein, auf die Wahl der Form des Absorptionskoefftzienten emphndlich reagieren.
AHHOTB~JI--IJ 06bIIqHOtinpaKTMKe
paWeTa paCIIpe/JeJIeHWI TeMIIepaTypbI M TenJIOO6MeHa IS OnTAqeCKN IIJIOTHOtiCpene IlpMMeHReTCR npn6nmKeHHe POCCeJIaHfia AJIH JIyWCTOrO nOTOK: Tenna, T.K. ero @lopMa yRo6Ha RJIR peruemfl ~H@#lepeHIwanbHbIx ypaBHeHHti B '43CTHbIX IIpOH3BOfiHbIX. OnHaKO AaHHbIX 0 KOL'@jjI4~MeHTe a6cop6I&m, BXOAR~eM B aIInpOKCHMa~MI0. B BIlAeWIeHa,XapaKTepH3yIO~erO H(H~KOCTb,He~OCTaTO~HO.~ pe3yJIbTaTe aHaJIM3a BbIRCHMJIOCb, YTO B CJIyWe 06TeKaHMR IIJIOCKOt IIJI3CTHHbI OAHOpORHOti HCIIAKOCTbIO (OI1TB~eCliIl nnoTaoti)pac4eTmIe npo@am TemepaTypbI A TerIJIOO6MeHa Hanyqemebf Ha CT~HK~:~BHMC~~T OT Rbl6paHHOti ~OpMblKO3@I$lUI~lleHT3 a6cop6qm.